Index

Symbols

/ 3-D Pose

@ \\

A

angle between vectors

associative property augmented matrix

B

basis vectors, [1], [2] bidiag

bidiagonal decomposition bulge

C

Cardinality change of basis characteristic equation characteristic matrix chol chol_decomp Cholesky decomposition column space, [1], [2] column view of systems commutative property

computing eigenvectors cond condition number, [1], [2] convergence of eigenvalue algorithms convergence points covariance cr, [1] CR factorization cross cross product

D

deflation degree one Francis step degree two Francis step det, [1] determinant, [1], [2], [3] diag diagonal matrix

diagonalization difference equations dimension, [1] direct algorithm division dot dot product

E

Eckart–Young theorem economy SVD, [1] eig, [1], [2] eig22, [1] eigen-decomposition eigenpair eigenvalue eigenvalue properties eigenvector, [1] eigenvector properties

eigFrancis eigQR, [1] elementary matrices elimination elimination matrix elimination procedures eps Euclidean norm existence of the SVD theorem eye

F

Fibonacci sequence Francis algorithm, [1]

Francis step Frobenius norm full rank, [1]

G

Gaussian elimination, [1] generalized eigendecomposition geometric transforms

givens, [1] Givens rotation matrix Gram–Schmidt process

H

Hermitian matrix Hermitian transpose hess Hessenberg decomposition Hessenberg form hessForm

homogeneous transformation matrix homography householder, [1] Householder reflection Householder reflector Householder transform

I

identity matrix ill-conditioned matrix implicit Q theorem implicitly shifted QR independent eigenvectors theorem

independent vectors inner product inv, [1], [2] inverse, [1], [2] inverse of matrix invertible test

L

l_1 norm l_2 norm Laplace expansion least squares regression left null space linear equations

linear independence linear vector properties lower triangular matrix LR algorithm lu, [1] LU decomposition

M

Markov matrix matrix matrix 2-norm matrix factorization

matrix inverse, [1], [2] matrix multiplication matrix norm Moore-Penrose pseudo-inverse, [1]

N

nearly singular matrix norm, [1], [2] normalized vector null, [1], [2], [3], [4]

null space, [1] null space from the SVD numerical stability, [1] numpy numpy.linalg.solve

O

orth, [1], [2] Orthogonal basis vectors orthogonal matrix orthogonal matrix factoring orthogonal matrix subspaces

orthogonally equivalent matrices theorem orthonormal columns orthonormal inverse/transpose outer product over-determined over-determined systems

P

p-norm parametric form partial pivoting PCA pinv, [1] pivot poly polyfit polynomial roots

poorly condition matrix positive definite matrix power method powers of matrix product of eigenvalues corollary product of transposes projection, [1], [2] properly Hessenberg pseudo-inverse, [1]

Q

qd algorithm qr QR algorithm QR algorithm with implicit shifts QR decomposition

QR decomposition!economy QR factorization QR step qrFactor quasitriangular matrix QZ algorithm

R

rank, [1], [2], [3], [4], [5] rank from the singular values theorem Rayleigh quotient rcond rcond number real real eigenvalues of a symmetric matrix theorem reduced row echelon form, [1] reflection matrix right hand rule, [1]

right null space roots, [1] roots of a polynomial rotation matrix round-off errors row exchanges row space row view of systems RREF rref, [1], [2], [3], [4]

S

Schur decomposition Schur triangularization theorem schurFrancis scipy scipy.linalg.solve similar matrices theorem similarity transform singular singular matrix, [1] Singular Value Decomposition span, [1]

spectral decomposition spectral decomposition theorem spectral theorem subspace iteration sum of eigenvalues corollary SVD, [1] svd SVD!economy, [1] svd65 svdFrancis symmetric matrix, [1]

T

testing eigensystem correctness trace, [1], [2]

transformation matrix transpose triangular matrix

U

under-determined systems unit vector

unitary matrix upper triangular matrix UTeigVect, [1]

V

vecnorm vector, [1]

vector length vector norm vector space, [1]

W

Wilkinson shift, [1]

Z

zero matrix